Finite element numerical analysisof blood flow and temperature

distribution in three-dimensionalimage-based finger model

Ying HeDepartment of Modern Mechanics,

University of Science and Technology of China,Hefei, China and

Computational Biomechanics Unit, RIKEN, Saitama, Japan

Ryutaro HimenoComputational Biomechanics Unit, RIKEN, Saitama, Japan

Hao LiuComputational Biomechanics Unit, RIKEN, Saitama, Japan and

Faculty of Engineering, Chiba University, Chiba, Japan

Hideo YokotaComputational Biomechanics Unit, RIKEN, Saitama, Japan, and

Zhi Gang SunAdvanced Simulation Technology of Mechanics Co. Ltd, Saitama, Japan

Abstract

Purpose – The purpose of this paper is to investigate the blood flow and temperature distribution inhuman extremities.

Design/methodology/approach – The simulation is carried out from three aspects. Firstly, thehemodynamics in the human upper limb is analyzed by one-dimensional model for pulsalite flow in anelastic tube. Secondly, the blood flow and heat transfer through living tissues are described basing onporous media theory, and the tissue model is coupled with the one-dimensional blood flow model. Withrespect to geometric modeling, MR-image-based modeling method is employed to construct a realisticmodel of the human finger.

Findings – It is found that the temperature variation is closely related to the blood flow variation inthe fingertip and the blood flow distribution in the tissue is dependent on the locations of large arteriesand veins.

Originality/value – Blood flow and temperature distribution in a 3D realistic human finger arefirstly obtained by coupling the blood circulation and porous media model.

Keywords Flow, Porous materials, Finite element analysis, Blood, Limbs, Temperature

Paper type Research paper

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0961-5539.htm

The first author would like to thank Ms J. Sunaga and Mr N. Kakusho for the support with takingMR images. The work was supported in part by KAKENHI of JSPS, Research (C), No. 18560218.

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Received 14 May 2007Revised 28 February 2008Accepted 28 February 2008

International Journal of NumericalMethods for Heat & Fluid FlowVol. 18 No. 7/8, 2008pp. 932-953q Emerald Group Publishing Limited0961-5539DOI 10.1108/09615530810899033

1. IntroductionThe human finger has a high-sensory capability that can convey information aboutmechanical, thermal, and tissue-damaging events. It has been known that the fingerskin temperature is closely related to the total digital blood flow; for example, aftercigarette smoking, peripheral circulation degrades and skin temperature decreases(Bornmyr and Svensson, 1991). When people are in a state of stress, the finger skintemperature decreases for the vasoconstriction of finger arterioles (Nketia andReisman, 1997). While warming distant torso body areas, blood circulation to thefingers is augmented and temperature of fingers increases to a comfortable levelduring exposure to cold (Koscheyev et al., 2000). Owing to the human finger’shigh-sensory capability, it has potential applications in developing new health careinstruments. For instance, Cho et al. (2004) proposed a noninvasive method formeasuring glucose, where glucose was derived by measuring heat generation, bloodflow rate, and hemoglobin oxygenation in a person’s fingertip.

Owing to the unique characteristics in hemodynamics and thermoregulation andpotential applications, developing a computational model of the human extremity orhuman body that can simulate the physiological characteristics is always a subject forfurther investigation.

Yokoyama and Ogino (1985) put forward a computer model to investigatecold-stressed effects on the finger. They developed a physiological model of the fingeraccording to measurements and anatomical structure. The finger was modeled in theshape of a cylinder comprising several layers, including bone, tendon, and skin. Theheat transfer process was depicted by three equations: those for heat transportprocesses in tissue, arterial, and venous pools.

Ling et al. (1995) proposed a finite element (FE) model to investigate the coolingeffects of water bags on a human leg. The modeled leg comprised skin, fat, bones, andtwo main arteries. The Pennes bio-heat equation was used, and the blood perfusionterm was assumed to be a function of the local tissue temperature. They considered theblood flow inside main arteries as non-Newtonian laminar flow.

Shitzer et al. (1997) presented a thermal model of an extremity such as a finger. Thismodel includes not only the effects of heat conduction, metabolic heat generation, andheat transport by blood perfusion, but also heat exchange between tissue and largerblood vessels and arterio-venous heat exchange. The heat transfer coefficient for majorblood vessels is derived from the relationship between the blood vessel radius,“influence volume” radius, cylinder radius, and thermal conductivity.

Researchers also tried to investigate global thermoregulation behavior byconstructing a global thermoregulation model. Takemori et al. (1995) proposed athermal model for predicting human comfort in an indoor environment. In their model,the human body was modeled into several cylinders and every cylinder was connectedby a circulatory system. The respiratory system was also considered in their model.Salloum et al. (2005) provided a similar thermal model that included 15 cylindricalparts and whose circulatory system was considered in greater detail. In each bodysegment, the blood exiting from the arteries was divided into blood flowing into thecore and blood flowing into the skin layer. The opposite action occurred in the veins ofthe same segment. The model was used to predict the nude body temperature duringdifferent ambient conditions and activities.

Finite elementnumerical

analysis

933

The common characteristics of the above models are that they used one or severalcylinders to model the geometry of the human body and their governing equationswere developed from the Pennes (1948) equation, but the hemodynamics of thecirculation system was still not considered.

On the other hand, a blood-perfused biological tissue can be described as a porousmedia in which the fluid phase represents the blood and the surrounding tissue isrepresented by the solid phase. The theory of porous media for heat transfer in livingtissues may be the most appropriate since it can describe the perfused blood with fewerassumptions as compared to other bioheat models (Khaled and Vafai, 2003). Wulff(1974) first dealt with the living tissue as a porous medium and utilized the convectiveterm, including the Darcy velocity, to replace the blood perfusion term in the bioheatequation. Xuan and Roetzel (1997) used the transport theory through porous media tomodel a tissue-blood system and analyzed the temperature distribution in humanupper limbs. Blood and tissue were considered to be in a nonequilibrium state, and twoenergy equations were used to express heat transfer in the blood phase and solid phase.The advantage of this model is that it includes the exact blood perfusion in tissues,blood dispersion, and effective tissue conductivity, and it is considered to beappropriate for modeling a blood-perfused tissue with fewer assumptions. However,the flow in large blood vessels differs from that through tissues and may be consideredseparately.

Besides, modeling heat ransfer in living tissue, modeling the detailedcountercurrent microvascular network is also of significance. Chen and Homles(1980) treated the blood vessels as two groups in their model – large and smallvessels. Each vessel is treated separately in the former group, whereas all vessels aretreated as a part of a continuum in the latter group. The thermal contributions of thesmall blood vessels were considered from the equilibration of blood temperature,convection of the flowing blood, and the small temperature fluctuations of the nearlyequilibrated blood. Weibaum and Jiji (1985) proposed an alternative model thataccounts for the thermal effect of the directionality of the blood vessels and thecharacteristic geometry of the blood vessel arrangement. The vascular structure in theperiphery was treated individually rather than as continuum media in theirthree-layer model.

One-dimensional nonlinear fluid model has been widely used to simulate the humanblood circulation system (arteries, capillaries, and veins). For example, a globalmathematical model of cerebral circulation in humans presented by Zagzoule andMarc-Vergnes (1986) a computational model to investigate the behavior of the entirehuman systemic circulation developed by Sheng et al. (1995) and structured-treesystemic arterial model presented by Olufsen et al. (2000). Applying the hemodynamicmodel to the thermal analysis of blood blow may be an alternative method. He et al.(2004) focused their attention on the heat transfer of blood flow by developing aone-dimensional thermifluid model.

With the development of image processing and mesh generation technique, usingrealistic geometric computational models to analyze the biomechanical characteristicsis becoming a tendency. Fukuda et al. (2005) investigated the deformation of the softtissue in the fingertip using a 3D realistic computational model constructed from CTimages. Reynolds et al. (2006) have constructed an accurate geometric model of theforearm and hand muscles and they intend to apply the model to analyze the

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mechanical motions of the forearm and the hand, along with the associated musclemovement. He et al. (2006) concentrated their efforts in this area by using a realistic 2Dfinger model constructed from MR images to investigate the blood flow andtemperature distribution based on the transport theory in porous media.

The purpose of this study is to model blood-tissue heat transfer according to thedifferent characteristics of blood flow in large vessels and tissues from three aspects.First, systemic blood circulation in the upper limb has been modeled based onone-dimensional flow in an elastic tube. Second, the 3D FE model based on heattransport in porous media has been developed to analyze the blood perfusion and heattransport in the human finger. Further, a realistic 3D geometric model for the humanfinger has been constructed on the basis of MR image data. By coupling theone-dimensional blood flow model and the porous media model, the blood pressure,velocity, and temperature of blood and tissues in a human finger were simulated.

2. Modeling blood flow dynamics and heat transfer in tissuesThe basic concept in modeling blood flow is the use of different models to simulateblood flow in large vessels (with diameters greater than or equal to 1 mm) and inmicrovessels. A nonlinear one-dimensional thermo-fluid model in an elastic tube is usedto express blood flow in large vessels, and blood flow perfused in tissues is consideredas the fluid phase in porous media. The energy equation for porous media is used tomodel the heat transfer in tissues.

2.1 Blood flow dynamicsThe blood flow in large vessels has been modeled as one-dimensional flow in an elastictube, and the governing equations, including continuity and momentum, are expressedas:

›A

›tþ

›q

›x¼ 0; ð1Þ

›q

›tþ

›

›x

q2

A

� �þ

A

r

›P

›x¼ 2

2pnr

d

q

A; ð2Þ

where x is the distance from the heart; t, the time; A, the cross-sectional area of theblood vessel; q, the blood flow rate; P, the transmural pressure; r, the blood density; n,the kinematic viscosity; d, the boundary-layer thickness; and r, the radius of the bloodvessel.

The pressure-area relationship for the arteries and veins is as follows:

Pðx; tÞ2 P0 ¼4

3

Eh

r01 2

ffiffiffiffiffiffiA0

A

r !; ð3Þ

p2 p0 ¼ kp 1 2A

A0

� �23=2" #

; ð4Þ

Finite elementnumerical

analysis

935

where E is Young’s modulus; h, the wall thickness of the blood vessel; and kp, thecoefficient proportional to the bending stiffness of the tube wall.

The energy equation for the one-dimensional model is written as:

›Tb

›tþ

q

A

›Tb

›x¼ 2vTb 2

hvesAs

rbcbAðTb 2 T tmnÞ: ð5Þ

A periodic flow wave was assigned at the inlet boundary, and a constant pressure wasassigned at the outlet. With regard to the bifurcation conditions and the junctionconditions between two equivalent tubes, it was assumed that there is no leakage ofblood at the bifurcations, the inflow and outflow are balanced, and the pressure iscontinuous.

The two-step Lax-wendroff finite difference scheme has been broadly used tocompute the blood flow and cross-sectional area for the interior points of each segmentof vessels in one-dimensional hemodynamic models (Zagzoule and Marc-Vergnes,1986; Sheng et al., 1995; Olufsen et al., 2000; Li and Cheng, 1993). Combining boundaryconditions for the pressure and flow rate at the bifurcation with this scheme, theparameters at the bifurcation points can be computed (Zagzoule and Marc-Vergnes,1986). The same scheme as the work of Zagzoule and Marc-Vergnes (1986) was used tosolve the set of nonlinear equations (He et al., 2004). The computed pressure, velocityand temperature of the corresponded vessels in the finger were transferred to the FEtissue model and assigned to the nodes represented as the blood. Thus, effect of largevessels can be added in the FE tissue model.

2.2 Darcy model and energy equation for biological tissuesThe Darcy model is considered to be the earliest flow transport model in porous mediaand is expressed as:

7P ¼ 2m

Kn; ð6Þ

where K is the permeability of the tissues; m, the viscosity; and n, the Darcy velocity.Based on the assumption that there is a local thermal equilibrium between solid tissueand blood flow, the energy equation is expressed as:

½ð1 2 wÞðrcpÞs þ wðrcpÞf �›Tm

›tþ ðrcpÞf n ·7Tm

¼ 7 · ð½ð1 2 wÞks þ wkf �7TmÞ þ ½ð1 2 wÞqs þ wqf �; ð7Þ

where:

ðrcÞm ¼ wðrcÞb þ ð1 2 wÞðrcÞt ð8Þ

km ¼ wkb þ ð1 2 wÞkt; ð9Þ

qm ¼ wqb þ ð1 2 wÞqt; ð10Þ

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are the overall heat capacity per unit volume, overall thermal conductivity, and overallheat production per unit volume of the tissue, respectively, and f is the porosity of thetissue.

Considering the continuity equation and momentum equation, the dimensionlesspressure in porous media is written as:

›2P*

›x*2þ

›2P*

›y*2þ

›2P*

›z*2¼ 0: ð11Þ

The dimensionless velocity is expressed as:

u* ¼ 2DaRe›P*

›x*v* ¼ 2DaRe

›P*

›y*w* ¼ 2DaRe

›P*

›z*; ð12Þ

where Da is the Darcy number and is expressed as:

Da ¼K

D 2: ð13Þ

The dimensionless energy equation is as follows:

›T*

›t*þ 1 u*

›T*

›x*þ v*

›T*

›y*þ w*

›T*

›z*

� �¼

1

Pem

›2T*

›x*2þ

›2T*

›y*2þ

›2T*

›z*2

� �

þ1

Pemq*m;

ð14Þ

where Pem, 1, and q*m are expressed as follows:

Pem ¼U1D

am; ð15Þ

1 ¼ðrcÞbðrcÞm

; ð16Þ

q*m ¼qmD

2

ðTa 2 T1Þkm: ð17Þ

In the air, the heat rate conducted to the skin surface is equal to the heat rate to the airfrom the surface by convection, radiation and evaporation. Therefore, the boundarycondition can be expressed as:

2ls›T

›n

����G

¼ hcðT* 2 T*1Þ���Gþhra ðT* 2 T*1Þ

���GþE*sk: ð18Þ

The dimensionless equation is written as:

Finite elementnumerical

analysis

937

2›T

›n

����G

¼ BiTjGþhraD

lsTjGþEsk; ð19Þ

where the parameter Esk is expressed by:

Esk ¼E*skD

lðT*a 2 T*1Þ: ð20Þ

When the finger is in the moving air whose velocity is in the range of0 , V , 0.15 m/s, the heat transfer coefficient at the surface of the finger is4.0 W/m2K. With respect to the radiative heat transfer coefficient, we used the constantvalue of 4.7 W/m2K as representative for the typical indoor temperature (ASHRAE,1997).

Evaporative heat loss E*sk from skin depends on the difference between water vaporpressure for the skin and for the air (Clark and Edholm, 1985). It can be written as:

E*sk ¼ he P*s 2 P*amb

� �: ð21Þ

The evaporative coefficient can be expressed as a function of air movement (Clark andEdholm, 1985) by:

he ¼ 124

ffiffiffiffiffiffiffiffiffiffiffiV*

amb

qW=m2kPa: ð22Þ

Equations (11)-(12) and (14) have been discretized using the Galerkin weighted residualmethod, and the conjugate gradient method was employed to solve the equations. First,the capillary pressure in tissues was computed and the value of the blood pressure inthe large vessels was obtained from the one-dimensional model and was assigned asthe boundary condition of equation (11). Second, the blood flow velocities in the tissueswere computed and the slip condition was employed at a large vessel wall. Finally, thetemperatures in large vessels and tissues were computed. A constant bloodtemperature condition was assigned to the inlet of the large arteries in the finger. Theflowchart of the computational method is shown in Figure 1.

3. Geometrical modelling for human finger from MR imagesSince we used a one-dimensional flow model to describe blood flow in arteries andveins, data of the cross-sectional area and the length of the blood vessel are requiredand are defined on the basis of previous works (Sheng et al., 1995; Olufsen et al., 2000).

3.1 MR image acquisitionThe original images were acquired for a volunteer’s finger. A hand-fitted supportermade of silicon rubber was produced in order to fix the volunteer’s hand before takingimages. A 1.5-T scanner (Excelart, Toshiba Medical Systems) was used with differentsequences for taking images of blood flow and different tissues. The image resolutionwas 256 £ 256, and the slice thickness was set to 1.0 mm. A circular coil whosediameter fit the length of the volunteer’s finger was employed. Figure 2 shows theoriginal MR cross-sectional images in different positions.

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Figure 1.Flowchart of

computational method

Two-step Lax-WendroffMethod

Continuity,Momentum, and State

Equation

FE equations for bloodand Tissue Temperatures

Pressures andVelocities in largefinger arteries and

veins

Blood Circulation Model

FE Thermofluid Model of Solid Tissue

CG (Conjugate Gradient) method

Velocities of Blood inTissues

Pressure Distribution ofblood in tissues

Figure 2.Original MR

cross-sectional images ofhuman finger in different

positions

(a) Slice- No. 39

(b) Slice-No. 95

Finite elementnumerical

analysis

939

3.2 Edge detectionSeveral image processing operators were applied in turn to reduce the noise present inthe original image and to subtract the image data from the background. The imageprocessing was carried out by an image processing and analysis program – ScionImage (www.scioncorp.com/), which supports many standard image processingfunctions. As regards the edge detection of the middle finger, the images of otherfingers were firstly cut-off. Furthermore, smoothing, contrast enhancement, andarithmetic subtraction for the images of the middle finger were performed and theimages only with the information of the objected finger were thus acquired. Figure 3shows the processed images of the middle finger. These images were further processedmanually and only the bone and skin area were segmented, whose boundaries wereindicated by discrete pixels with different brightness. Figure 4 shows the finalprocessed images corresponding to Figure 3. The processed images can be exported astext files with information about the brightness of each pixel in the images. Table Ishows a part of the text file of the processed image in which the numbers represent thebrightness of every pixel. The area where the brightness number was not 254represents the finger area. The edge of the finger and the bone were set in severalspecial values of brightness; hence, the position of the skin and the bone of the humanfinger can be identified by detecting these determined pixels.

Figure 3.MR cross-sectional imagesof human finger indifferent positions afterimage processing

(a) Slice No. 39 (b) Slice No. 95

Figure 4.The images after theimage segmentation

254 254 254 80254 254 254 210254 254 210 210254 20 210 210254 210 210 210254 210 210 210254 210 210 21020 210 210 210210 210 210 210210 210 210 21020 210 210 210

Table I.Text file of processed MRimage

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3.3 Mesh generationFigure 5 shows the flow chart for the mesh generation. The text files with the imageinformation were subsequently input into the mesh generation program. Since the textfiles include the color information to express different materials, for instance, the colorindices 20, 70, 120, 80 express four segments of the skin boundary and the color indices100, 30, 40, 10, 90 reveal the four segments of the bone boundary, we can identify theseboundaries according to the edge information. Through this boundary information, thefinger domain was subdivided into three subregions, namely, bone, tendon, and skinand muscle subdomains. Since the characteristic color indices were used to definedifferent segments of every subdomain, the corresponding characteristic pixels on apair of opposite sides may be connected by straight lines, and the interior grid pointswere generated by linear interpolation.

Grids were firstly generated on the subdomain of bone and the nodes in theinterconnection with tendon were recorded in advance. Secondly, the same procedurewas conducted on the tendon subdomain by using the nodes in the interconnection asthe starting points. The nodes related to the boundaries of the two subregions wererecorded as the inner side of the muscle region. Furthermore, the nodes in the inner sideand the opposite skin side were connected by straight lines and unidirectionalinterpolation technique was employed for the grid generation in the third subregion.Figure 6(a)-(c) shows the 2D mesh from different processed cross-sectional images.After elements in each slice were generated, they were stacked to construct a 3D mesh.In this process, 24 slices were employed and the distance between two adjacent sliceswas 4 mm.

Figure 7 is result of the 3D mesh generation of the human finger. The main programfor mesh generation is listed in Appendix.

Figure 5.The Flowchart for meshgeneration from medical

images

Image Processing

Image J

Read The OriginalSequential Images andExport in correct order

Scion Image

Scion Image

Export the Text File of theImages

Original Code

Export the necessary Files forSimulation Computing

Identify Different Materials

Generate Hexahedral Elements

Read the Sequential Text File and GenerateTwo-dimensional Meshes According to the

Original Data

Finite elementnumerical

analysis

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3.4 Identification of blood vesselsRecognizing the position of a large blood vessel in the MR images of the finger isslightly difficult due to the small size of the vessels. Hence, we first chose MR imagesfor the base of the finger, where the diameter of the vessel is larger and easier to judge.Next, the position of the center of the arteries was detected and the information wasadded to the corresponding four-node FE model. Furthermore, the relative position ofthe arteries with respect to the edge of the bone in the FE model was recorded, and thesame relative position of the arteries was used in the other four-node FE models. Thus,the 3D FE network of the arteries was generated. It was assumed that the veins were inthe neighborhood of the arteries and that they had the same diameter as that of thearteries. We also assumed that blood vessels with a diameter smaller than that of thelarge artery were the fluid phase in the porous media.

4. Computation resultsThe thermophysical properties are listed in Table II; these properties were obtainedfrom references (Shitzer et al., 1997; Shih et al., 2003). It was considered that tissue

Figure 6.Two-dimensional FEmeshes generated fromMR cross-sectional images

Finger-slice-27

(a) Slice-No. 27 (b) Slice-No. 39 (c) Slice-No. 95

Finger-slice-39 Finger-slice-95

Figure 7.3D FE network of thehuman finger

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porosity does not normally exceed 0.6 (Nield and Bejan, 1998). Based on the data inNield and Bejan (1998), we defined the porosity in the bone, tendon, skin, and fingertipas 0, 0.1, 0.2, and 0.3, respectively. Owing to the large density of the vessels at thefingertip, the porosity was assigned a larger value. Tissue permeability was defined as10213 m2 (Nield and Bejan, 1998).

The computational code was validated with several simple shapes. Figure 8(a)shows a long narrow bar and Figure 8(b) shows the temperature distribution in the barwhen a given temperature boundary condition and a given heat transfer coefficientwere assigned on two opposite sides. Another validation was carried out in the shapeas Figure 9(a), where two small tubes were embedded in a solid. This shape can beconsidered as one part of living tissue with two countercurrent vessels. The computedcapillary pressure distribution in the tissue is shown in Figure 9(b).

Figure 10 shows the variation of pressure in the large arteries and veins in thefinger, and it was computed by the one-dimensional blood flow model. It can beobserved that the pressure difference is greater in the artery, whereas there is only aslight difference in the vein. These data were given to nodes that represent the arteriesand veins in the FE model and the pressure was set such that it varied along the bloodflow direction but did not change in the direction normal to the flow direction. Theinflow velocities in the finger artery and vein are 19 and 3.5 cm/s, respectively,according to the computation results obtained from the one-dimensional model, and theReynolds number in the computation is 50 when the arterial velocity is the reference

Bone Tendon Skin Blood

r (kg/m3) 1,418 1,270 1,200 1,100c ( J/kgK) 2,094 3,768 3,391 3,300l (W/mK) 2.21 0.35 0.37 0.50

Table II.Physical properties andblood perfusion rates of

tissues

Figure 8.(a) The long narrow bar

for the validation;(b) temperature

distribution in the bar

1.00

0.95

0.91

0.86

0.82

0.77

(a) (b)

Finite elementnumerical

analysis

943

velocity. Since the computed velocity along the flow direction slightly changes in largevessels, it is assigned to the blood nodes with uniform values.

Figure 11(a)-(b) shows the computed capillary pressure of the finger in differentcross-sectional areas. It can be seen that the capillary pressure is between the values ofpressure in large artery and vein. Figure 12(a)-(b) shows the flow distribution of thefinger in different position. It can be seen that blood flows in the tissue between thelarge arteries and veins; however, there is almost no blood flow in the tissue betweenthe two large arteries. Furthermore, the blood flow in the fingertip is greater than thatin the basal part. In addition, in some areas, the blood flow is in the direction normal tothe skin surface, and in other areas, the blood flow is parallel to the skin surface.Therefore, we can identify the thermally significant and insignificant vessels easily.

Figure 9.(a) A test cube with twosmall vessels; (b) theimages after the imagesegmentation

200

162

125

87

50

12

(a) (b)

Figure 10.Pressure distribution inaxial direction of bloodflow in large arteries andveins of finger

0 20 40 600

20

40

60

80

100

Blo

od P

ress

ure,

mm

Hg

Vessel Length, mm

Pressure in the large arteries of the Finger

Pressure in the Large veins of the Finger

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The flow distribution also shows agreement with the anatomical structure ofmicrovessels.

Figure 13 shows the temperature distribution in the cross-sectional areas of thefinger when it was exposed to air. Convective heat transfer can be observed betweenlarge arteries and veins. In the areas near the large vessels where there are morethermally significant capillaries, the tissue temperature is distinctly higher. In thefingertip, the tissue and skin surface temperature are higher due to more blood flow.

Figure 14 shows the skin temperature distribution when the blood is perfused andnot perfused in the tissue. It can be seen that the blood affects the skin temperature inthe fingertip more obviously, which means that blood plays a more important rolethermally at the fingertip.

Figure 15(a) shows the skin temperature profile from the fingertip to the basal partof the finger in two different places. It can be seen that the temperature at the fingertipis generally higher than that in the basal part of the finger, and there are fluctuations inthe two profiles. Figure 15(b) and (c) shows the measured skin temperature profiles

Figure 11.Capillary pressure

distribution in differentpositions

280

Z = 7.5 mm Z = 80 mm

235

189

144

98

52

280

235

189

144

98

52

Figure 12.Blood flow distribution in

tissue in differentcross-sectional areas

z = 7.5 mm z = 80 mm

0.00100 0.00100

Finite elementnumerical

analysis

945

Figure 13.Temperature distributionin different cross-sectionalareas of finger while it isexposed to air

1.0

z = 6 mm

0.9

0.8

0.8

0.7

0.6

1.0

z = 19 mm

To

the

fing

ertip

0.9

0.8

0.8

0.7

0.6

1.0

z = 80 mm

0.9

0.8

0.8

0.7

0.6

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946

from the fingertip to the basal part for different subjects. It can be seen that thevariation tendency of the simulated results is in agreement with that of themeasurements.

5. Discussion and conclusionsIn this study, we divided the blood flow in the tissues into two groups: one is blood flowin large vessels (the diameter of the vessels is generally larger than or equal to 1 mm)and the other is blood flow in microvessels. We used a one-dimensional thermofluidmodel in an elastic tube to describe the blood flow in large vessels and the Darcy modelto model the flow in porous media to describe the blood flow in microvessels. Theenergy equation in porous media was used to model the heat transfer in the tissue. Thecomputed results have shown that the simulated blood flow through tissues isqualitatively in agreement with the anatomical structure and the measurement. Sinceinformation on the local blood flow has been obtained from the one-dimensional flowand Darcy models, the thermal effect of different vessels can be evaluated through theenergy equation without other parameters.

Another characteristic of this study is that the image-based modeling method wasemployed to construct the FE model. Since the realistic geometry can be constructedfrom the medical images directly, we can simulate the blood flow and heat transfer in amore realistic condition. This simulation system can be extended to apply it toanalyzing other heat transfer problems in other living tissue.

One limitation of the computational system is that it cannot simulate heat transferphenomena in the transient state because we assumed that the blood temperature inthe tissue achieved equilibrium with the tissue temperature. There is a need to considerthe blood temperature and tissue temperature by separate equations so that transientbioheat transfer phenomena can be simulated.

In summary, the simulated system including the one-dimensional hemodynamicmodeling, the heat transfer modeling basing on the porous media theory, andimage-based geometric modeling provides favorable results in analyzing blood flowand heat transfer problems in living tissues. It is believed that based on a similarconcept, the nervous system can also be modeled and more challenging problems withregard to human physiological behavior can be explored.

Figure 14.Skin temperature

distribution for differentblood perfusions

1.0

Dimensionless TemperatureWith blood perfusion

Dimensionless TemperatureWithout blood perfusion

0.9

0.8

0.8

0.7

0.6

1.0

0.9

0.8

0.8

0.7

0.6

Finite elementnumerical

analysis

947

Figure 15.(a) Simulated skintemperature profile fromthe fingertip to the basalpart; (b) measured skintemperature profile fromthe fingertip to the basalpart for subject 1 andsubject 2

0.6

0.7

0.8

0.9

Node Number from the Fingertip to the Basal Part(a)

(b)

Dim

ensi

onle

ss T

empe

ratu

re, T

*

5,000 10,000

Location 1

Location 2

32.5

33

33.5

34

34.5

35

35.5

33.534

34.535

35.536

36.5

1 4 7 10 13 16

Pixel from the Fingertip to theBasal Part of the Finger

Pixel from the Fingertip to theBasal Part of the Finger

Tem

pera

ture

Tem

pera

ture

19 22 25 28 31 34

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

1

50.1°C

45.140.135.130.125.120.1

10.115.1

50.1°C

45.140.135.130.125.120.1

10.115.1

HFF18,7/8

948

References

ASHRAE (1997), ASHRAE Fundamentals Handbook, ASHRAE Press, Atlanta, GA.

Bornmyr, S. and Svensson, H. (1991), “Thermography and laser-Doppler flowmetry formonitoring changes in finger skin blood flow upon cigarette smoking”, Clinical Physiology,Vol. 11, pp. 135-41.

Chen, M.M. and Homles, K.R. (1980), “Microvascular contributions in tissue heat transfer”,Annals of the New York Academy of Sciences, Vol. 335, pp. 137-50.

Cho, O.K., Kim, Y.O., Mitsumaki, H. and Kuwa, K. (2004), “Noninvasive measurement of glucoseby metabolic heat conformation method”, Clinical Chemistry, Vol. 50 No. 10, pp. 1894-8.

Clark, R.P. and Edholm, O.G. (1985), Man and His Thermal Environment, Edward Arnold(Publishers) Ltd, London.

Fukuda, S., Sakai, N. and Shimawaki, S. (2005), “Numerical simulation of mechanicaldeformation of three-dimensional human fingertip model constructed from CT data”,Proceedings of the 16th JSME Conference on Frontiers in Bioengineering, Kusatsu,November 10-11.

He, Y., Liu, H. and Himeno, R. (2004), “A one-dimensional thermo-fluid model of blood circulationin upper limb of man”, International Journal of Heat and Mass Transfer, Vol. 47 Nos 12/13,pp. 2735-45.

He, Y., Liu, H., Himeno, R., Sunaga, J., Kakusho, N. and Yokota, H. (2006), “Finite elementanalysis of blood flow and heat transport in the human finger”, International HeatTransfer Conference IHTC-13, Sydney, August.

Khaled, A.R.A. and Vafai, K. (2003), “The role of porous media in modeling flow and heattransfer in biological tissues”, Int. J. Heat Mass Transfer, Vol. 46, pp. 4989-5003.

Koscheyev, V.S., Leon, G.R., Paul, S., Tanchida, D. and Linder, I.V. (2000), “Augmentation ofblood circulation to the fingers by warming distant body areas”, European Journal ofApplied Physiology, Vol. 82, pp. 103-11.

Li, C.W. and Cheng, H.D. (1993), “A nonlinear fluid model for pulmonary blood circulation”,J. Biomechanics, Vol. 26 No. 6, pp. 653-64.

Ling, J.X., Young, I.R. and DeMeester, G. (1995), “A finite element model for determining thecooling effects of water bags on the temperature distributions of a human leg”, Advancesin Heat andMass Transfer in Biotechnology, ASME, HTD-Vol. 322/BED-Vol. 32, pp. 69-71.

Nield, D.A. and Bejan, A. (1998), Convection in Porous Media, 2nd ed., Springer PublishingCompany, New York, NY.

Nketia, P. and Reisman, S. (1997), “The relationship between thermoregulatory andhemodynamic responses of the skin to relaxation and stress”, Proceedings of the IEEE,23rd Northwest Bioengineering Conference, IEEE, New York, NY, pp. 27-8.

Olufsen, M.S., Peskin, C.S., Kim, W.Y., Pedersen, E.R., Nadim, A. and Larsen, J. (2000),“Numerical simulation and experimental validation of blood flow in arteries withstructured-tree outflow conditions”, Annals of Biomedical Engineering, Vol. 28, pp. 1281-99.

Pennes, H.H. (1948), “Analysis of tissue and arterial blood temperatures in the resting humanforearm”, Journal of Applied Physiology, Vol. 1 No. 2, pp. 93-122.

Reynolds, H.M., Smith, N.P. and Hunter, P.J. (2006), “Construction of an anatomically accurategeometric model of the forearm and hand musculo-skeletal system”, Proceedings of the26th Annual International Conference of the IEEE EMBS, San Francisco, CA, September,pp. 1829-32.

Salloum, M., Ghaddar, N. and Ghali, K. (2005), “A new, transient bio-heat model of the humanbody”, Proceedings of HT2005, 2005 ASME Summer Heat Transfer Conference.

Finite elementnumerical

analysis

949

Sheng, C., Sarwal, S.N., Watts, K.C. and Marble, A.E. (1995), “Computational simulation of bloodflow in human systemic circulation incorporating an external force field”, Medical &Biological Engineering & Computing, Vol. 33, pp. 8-17.

Shih, T.C., Kou, H.S. and Lin, W.L. (2003), “The impact of thermally significant blood vessels inperfused tumor tissue on thermal dose distributions during thermal therapies”, Int. Comm.Heat Mass Transfer, Vol. 30 No. 7, pp. 975-85.

Shitzer, A., Stroschein, L.A., Vital, P., Gonzalez, R.R. and Pandolf, K.B. (1997), “Numericalanalysis of an extremity in a cold environment including countercurrent arterio-venousheat exchange”, Trans. ASME, Journal of Biomechanical Engineering, Vol. 119, pp. 179-86.

Takemori, T., Nakajima, T. and Shoji, Y. (1995), “A fundamental model of the human thermalsystem for prediction of thermal comfort”, Trans. JSME, Vol. 61 No. 584, pp. 1513-20.

Weibaum, S. and Jiji, L.M. (1985), “A new simplified bioheat equation for the effect of blood flowon local average tissue temperature”, J. Biomech. Eng., Vol. 107, pp. 131-9.

Wulff, W. (1974), “The energy conservation equation for living tissue”, IEEE Transactions onBiomedical Engineering, pp. 494-5.

Xuan, Y.M. and Roetzel, W. (1997), “Bioequation of the human thermal system”, Chem. Eng.Technol., Vol. 20, pp. 269-76.

Yokoyama, S. and Ogino, H. (1985), “Developing computer model for analysis of human coldtolerance”, The Annals of Physiological Anthropology, Vol. 4 No. 2, pp. 183-7.

Zagzoule, M. and Marc-Vergnes, J. (1986), “A global mathematical model of the cerebralcirculation in man”, J. Biomechanics, Vol. 19 No. 12, pp. 1015-22.

(The Appendix Figure follows overleaf.)

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Appendix. The original main program for generating brick element

(continued)Figure A1.

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(continued)Figure A1.

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Corresponding authorYing He can be contacted at: yhe@ustc.edu.cn

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Figure A1.

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